I'm reading about amenable groups. What are explicit examples of nonabelian discrete amenable groups other than finite groups? Perhaps a group presentation or matrix representation would be useful.
9 Answers
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Solvable (which includes nilpotent and polycyclic); locally finite; subexponential growth.
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$\begingroup$ Infinite Dihedral Group falls under this one. $\endgroup$ Sep 26, 2021 at 20:18
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To "specify" Alain's answer: 0) The group $\langle a,b \mid bab^{-1}=a^2\rangle$ (solvable of class 2 Baumslag-Solitar group) 1) The group of upper triangular $n\times n$ matrices with integer coefficients and 1 on the diagonal (nilpotent), $n\ge 1$. 2) The group of all permutations of $\mathbb{Z}$ with finite support (locally finite). 3) The subexp. growth groups, unfortunately, would require more space to define. But you can find them in Wiki.
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The lamplighter group, defined as the wreath product $\mathbb{Z}/2\mathbb{Z} \wr \mathbb{Z}$, is amenable yet has exponential growth. It can be thought of as a bi-infinite sequence of street lamps, each of which can be turned on and off, and a lamplighter who moves along the sequence. The three generators of the group are to move the lamplighter right or left, and to switch the state of the lamp he is positioned in front of. With this picture in mind, it is easy to show the group has exponential growth.
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Groups generated by bounded automaton are amenable:
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Elementary amenable groups: the smallest class of groups which includes finite groups and commutative groups, and is closed under formation of homomorphic images, subgroups, group extensions (by an other element of the class) and directed unions. (See http://en.wikipedia.org/wiki/Elementary_amenable_group)
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$\mathrm{Symm}(\mathbb{Z}) \leftthreetimes \mathbb{Z}$ - see page 4321 in http://home.gwu.edu/~maxal/a-g-g.pdf
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Solvable and locally finite are elementary, I guess those of subexponential growth are, too.
E.g. a solvable group is one that can be constructed from abelian groups using extensions. But now I see that locally finite groups are not elementary (for example, the free group on two generators is finitely generated, hence locally finite, but it is definitely not amenable, hence it cannot be elementary), sorry for that.
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$\begingroup$ Free groups are not locally finite. I think you meant that inverse limit of finite groups need not be amenable. $\endgroup$– MishaApr 4, 2013 at 20:59
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1$\begingroup$ free groups are not inverse limits of finite groups neither (at least not in the category of groups). They are residually finite, anyway (that is, in the category of marked groups, the same as inverse limits of finite groups). $\endgroup$– YCorApr 4, 2013 at 22:01
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$\begingroup$ Yves, I meant that they embed in inverse limits of finite groups. $\endgroup$– MishaApr 4, 2013 at 23:03
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Although not directly concerned with the OP but still interesting to know (and since it is community-wiki): finitely generated amenable groups have finitely many ends (i.e. 0, 1 or 2), since amenable groups cannot contain non-abelian free groups (however there is also a direct argument proving this statement without using Stalling's End Theorem)